Independence stochastic integral

Question

I've been finding some difficulties in an exercise about the stochastic integral:

Consider the following stochastic process: $$X_t=\int_0^t\sigma_udW_u$$ Where $\sigma_u$ is a cadlag deterministic adapted process, $W_u$ is a standard Brownian motion and $\{\mathcal{F}_t\}_{t\geq 0}$ is the standard augmented filtration. Is it true that $X_t-X_s$ is independent on $\mathcal{F}_s$?

MY ATTEMPT

Note that the process $X_t-X_s=\int_s^t\sigma_udW_u$ and for any partition of the interval $[s,t]$ of the type $\pi^n=\{s=t_0^n<t_1^n<...<t^n_{k_n}=t\}$ with $\max|t^n_i-t^n_{i-1}|\xrightarrow[n\to\infty]{}0$: $$\lim_{n\to\infty}\sum_{i=1}^{k_n}\sigma_{t^n_{i-1}}(W_{t^n_i}-W_{t^n_{i-1}})\xrightarrow[n\to\infty]{\mathbb{P}}\int_s^t\sigma_udW_u$$ We have that for any $n$ the LHS is independent on $\mathcal{F}_s$ thanks to the independece properties of Brownian increments ($\sigma_s$ is deterministic).

Is there any result that allows me to say that also the limit is independent on $\mathcal{F}_s$?

Answer 1

Since the Itô integral is joint Gaussian it suffices to show that for $X_{t}=\int^t \sigma_r dW_{r}$

$$E[X_{s}(X_{t}-X_{s})]=0.$$

A càdlàg function is bounded on $[0,T]$ How to prove that càdlàg (RCLL) functions on $[0,1]$ are bounded?, and so you can just dominated convergence theorem to pass to the limit from the $L^2$-approximation with the dyadics partitions

$$E[X_{s}(X_{t}-X_{s})]=\lim_{n\to +\infty}E[X_{s}^{n}(X^{n}_{t}-X^{n}_{s})]=0.$$

Joint Gaussian

For $t^{n}_{k_{n}}=t^{n}_{k_{n}}(t)$ dyadic-approximation to $t>0$, the sums

$$\left(\sum_{i=1}^{k_{n}(t)}\sigma_{t^n_{i-1}}(W_{t^n_i}-W_{t^n_{i-1}})\right)_{t> 0}$$

form a joint-Gaussian process since sum of independent Gaussians is Gaussian eg. see here

Let $\{X_i\}_{i\in I}$ be a joint normal collection of normal random variables. Then, any collection $\{Y_j\}_{j\in J}$ of finite linear combinations of the $X_i$ is again joint normal.

Then as mentioned here The ito integral is gaussian, we simply take limits of their characteristic functions to get the Itô-integrals also having joint Gaussian.